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Statistical inference for density-dependent Markovian
forestry models
Stéphan Clémençon, Mélanie Zetlaoui
To cite this version:
Stéphan Clémençon, Mélanie Zetlaoui. Statistical inference for density-dependent Markovian forestry models. 2010. �hal-00484540�
Statistical inference for density dependent
Markovian forestry models
Abstract
A stochastic forestry model with a density-dependence structure is studied. The population evolves in discrete-time through stage-structured processes, in a way that its temporal evolution is described by a stochastic Markov chain. For adequate scalings of the transi-tion rates, it is shown to converge to the deterministic matrix model, known as the Usher model, as a parameter n, interpreted as the pop-ulation size roughly speaking, becomes large. From the perspective of the analysis of forestry data and predict the forestry population evo-lution, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as max-imum likelihood estimation. We state preliminary statistical results in this context. Eventually, relation of the model to the available data of a tropical rain forest in French Guiana is investigated and numerical applications are carried out.
Keywords. Population dynamics; matrix model; Markov chain; density dependence; large population approximation; maximum likelihood estima-tion.
1
Introduction
Models of population dynamics are an important tool in many ecological studies. They are used to mimic the future evolution of the population. Among the discrete-time models, matrix model are often used to study the dynamics of structured populations (either age-structured or size-structured populations). They also permit to simplify the dynamics of a population into its basic components: recruitment or birth, growth or ageing, and mortality. Matrix models have been widely used in ecology to deal with invasive species [18, 29], population viability [5, 3, 26, 28, 15], or the management of the harvested populations [13, 7, 16].
The most general matrix model was proposed by [21], and allows any transition from one stage to another. The Usher model [31, 32] is a size-structured population model and restricts the possible transitions: during on time step, an individual either stays alive in the same stage, moves up to
the next stage, or dies. It is often used in forestry because it is well adapted to the dynamics of a forest stand and it permits to simulate quickly and in a synthetic way large areas of forest [14]. This model is particularly adapted to deal with forest management [27, 8, 33, 4, 7, 19], economic potential of forests [25, 6], or biodiversity assessment [8, 25, 17, 24, 23]. The Leslie model [22] describes a population grouped by age, and is a special case of the Usher model: during each time step, an individual can only move up a class or die. Matrix models describe the dynamics of a population by the effectif vector which components are the number of individuals in each class. These models were at first deterministic. Demography stochasticity, consisting on describe the population dynamics by Markov chains, was used above all to deal with small-size populations, to model for instance the probability of extinction [5, 20, 26]. However some recent population modelling efforts have employed this type of modelisation [12, 9, 11, 1, 30].
In this paper we focuse on a generalized Usher model where individual evolutions are assumed to be dependent of the running density. This model take into account the tree interactions due to competition effects into in-dividual dynamicsAt any time, recruitment, growth and mortality in each state class depend on the overall density. Assuming a large population of tree, the properties of the mathematical model are investigated and prelim-inaries statistical questions are tackled. Beyond the stochastic modelling, the main goal is to establish large population limit results (law of large num-ber and limit central theorem) for the effective Markov chain describing the forest dynamics. This is a classical approach for the statistical analysis in population dynamics and its applications.
The paper is organized as follows. In section 1 a Markov chain with density-dependence in individual dynamics is introduced for modeling the temporal evolution of a forest stand. A short description giving an insight into how the dynamics is driven. In average, the evolution of the Markov chain verifies the deterministic relation given by the Usher model. A short probabilistic study is also given describing the long-term behavior of the Markov chain. The main part of this work is concerned by the section 3. Considering a Markov chain indexed by a positive natural n, representing approximatively the initial size of the population, limit results when n tends to infinity are established. These results provide maximum likelihood esti-mators that are consistent in a large population framework in section 4.
2
The markovian forestry model
2.1 The population dynamics
The population is grouped into I stages. Its evolution is described in discrete time by the random vector N (t) = (Ni(t))i=1...I, where Ni(t) is the number
in NI and is adapted with respect to the filtration F = (F
t)t∈N defined by :
Ft= σ (∪s≤tσ(N (s))). Between time t and t + 1 an individual either stays
in the same class, or grows up to the next stage, or dies (see Fig. 1). Then the model is written by:
Ni(t + 1) = Fi i(t) + Fi−1 i(t) (1)
where Fi j(t) is the flow of individuals from the stage i to the stage j between
time t et t + 1, with convention F0 1(t) = Rt where Rt being the number of
birth. We denote also Fi†(t) the number of individuals from the stage i
which die between the time t and t + 1. Flow law conditionally to N (t) is a multinomial law, and number of birth follows a Poisson’s law:
(Fi i(t), Fi i+1(t), Fi†(t)) ∼ M (Ni(t), pi(N (t)), qi(N (t)), mi(N (t))) Rt ∼ P( I X i=1 fi(N (t))Ni(t))
where pi(N (t)) is the transition rate for an individual in stage i which stays
in the same stage, qi(N (t)) the transition rate for an individual which grows
up to the next stage, and mi(N (t)) the death rate. Those parameters verify
the stochastic relation:
pi(N (t)) + qi(N (t)) + mi(N (t)) = 1 (2)
Furthermore fi(N (t)) denotes the fecundity of stage i.
With those assumptions, the average evolution of effective vector is given by the equation
E(N (t + 1)|N(t)) = U(N(t))N(t) (3)
where U is a matrix of size I× I, called "Usher" matrix. Its elements are
… …
1 2 3 i i+1 I
the transition parameters, that is U (N (t)) = P (N (t)) + R(N (t)) where: P (N (t)) = p1(N (t)) 0 · · · 0 q1(N (t)) p2(N (t)) ... . .. . .. 0 0 qI−1(N (t)) pI(N (t)) R(N (t)) = f1(N (t)) · · · fI(N (t)) 0 · · · 0 .. . ... 0 · · · 0
Example 1 (On modelling the density-dependence) Density dependence can take an infinity of shapes, but the most commonly found (see [9]) are the Berverton-Holt dependence:
v(N ) = c
b + N · a and the Ricker (or exponential) dependence:
v(N ) = c exp(−bN · a) (4)
where v is any of the vital rates, c and b are constant parameters, a is a constant positive vector of RI, and dot denotes the scalar product in RI.
For instance, a can be the unit vector (1, 1, ..., 1), so that N · a is the total number of individuals at time t. In forestry applications, a is often taken as the vector whose ith element is the average basal area of an individual in stage i. The quantity N · a then represents the cumulative basal area of the population. Exploratory analyses showed us that the exponential density-dependence better suited the experimental data at Paracou than the Beverton-Holt dependence. In what follows, we shall thus focus on the exponential dependence given by equation (4).
Nevertheless, this representation not assures the probability condition given by equation (2). For this reason, we propose the following model where each vital rate has his own density-dependence, that is to say:
pi(N ) = (1− γi) µieξiN·a µieξN·a+ νie−κiN·a (i = 1, . . . , I) qi(N ) = (1− γi) νie−κiN·a µieξiN·a+ νie−κiN·a (i = 1, . . . , I− 1) (5) f (N ) = αe−βN ·a
Especially, piincreases with N·a and tends to a constant 1−γiat infinity,
and not depends on the density. With this simplification, the mortality rate represents the natural death. To take into account competition in mortality, we propose the following shape for death rate:
mi(N ) = ηi+ δi(1− e−λiN·a) (6)
with ηi + δi < 1. The death rate is the sum of natural mortality, ηi, and
mortality due to competition. Also, when N· a tends to infinity, the death rate increases toward ηi+ δi.
2.2 Markov chain description
The sequence (N (t))t∈N that verifies equation (1) is a markov chain in NI
with initial law µ0 and transition probability denoted Π, defined, for all
m = (m1, . . . , mI) and n = (n1, . . . , nI) of NI, by : Π ((m1, . . . , mI), (n1, . . . , nI)) = I Y i=1 Πi(m, ni)
where Πi(m, ni) = P (Ni(t + 1) = ni|N(t) = m) is, for i ≥ 2, the convolution
of two binomial distributions, that is:
Πi(m, ni) = min(mXi,ni) k=ni−min(mi−1,ni) B1ik(p, m, n)Bik2(q, m, n), if ni ≤ mi+ mi−1 0 else
and, for i = 1, the convolution of a binomial and a Poisson distribution:
Π1(m, n1) = min(m1,n1) X k=0 B1k1 (p, m, n)Dk(f, m, n) where: Bik1(p, m, n) = Ck mipi(m) k(1− p i(m))mi−k for i = 1, . . . , I Bik2(q, m, n) = Cni−k mi−1qi−1(m) ni−k(1− q
i−1(m))mi−1−ni+k for i = 2, . . . , I
Dk(f, m, n) = PI j=1fj(m)mj n1−k (n1− k)! e−PIj=1fj(m)mj
2.3 Limiting behavior in long time asymptotics
We now state a limit result for the forestry markov chain, as time goes to infinity.
Proposition 1 Considering the markov chain (Nt)t>0 introduced in
defi-nition 1, we have, whatever the intitial state N0, that (Nt) tends to 0 in
distribution as t tends to infinity.
The details of the proof is given in Appendix A. This ergodicity result shows that the time of population extinction is almost surely finite, though it may be very long in practice. In situation of long-lasting dynamics, as a tropical forest, the long-term behavior can be refined by studying quasi-stationnarity measures.
3
Large population limit
3.1 Renormalization
We consider a sequence {(N(n)(t))t∈N; n ≥ 1} of Usher markov chain. For
n≥ 1, the markov chain (N(n)(t))
t∈N starting from N(n)(0), proportionnal
to n and with the modifications on transition rates v(n)(N(n)(t)) = v(N
(n)(t)
n ) (7)
where v verifies one of equations (5).
Remark 1 (On the meaning of the renormalization) With this renormaliza-tion, the competition term of any vital rate is function of the density. The definition of the competition then not depends of the plot area where the population lives.
Let consider a Usher markov chain starting from N(n)(0) during time T ,
(N(n)(0), . . . , N(n)(T )). If the population size increases, the term N(n)(t)/n
increases, and p(n)(N(n)(t)) increases while q(n)(N(n)(t)) decreases.
Let now eNn = N
n
n and T ∈ N
∗. Then eNn(T ) verifies the following
equation: e Nn(T ) = eN0n+ TX−1 s=0 F ( eNn(s)) + Mn(T ) (8) where:
• F is the function defined for all x = (xi)i=1,...,I in RI by:
• the series t−1 X s=0 F ( eNn(s)), t≥ 1 ! is Fn-predictable
• Mn= (Mn(t), t∈ N) is a Fn-martingale, defined for all t≥ 1 by:
Mn(t) = t−1 X s=0 h e Nn(s + 1)− ENen(s + 1)|Fn s i = t−1 X s=0 h e Nn(s + 1)− UNen(s)Nen(s)i and Mn(0) = 0.
3.2 The Law of Large Numbers
We want to show that eNn(T ) tends, when n tends to infinity and almost
surely, towards the deterministic vector yT in RI which verifies the following
equation:
yT = y0+ TX−1
s=0
F (ys) (9)
This proof is particularly based on the convergence to zero of the martingale Mn(t) when n tends to infinity and almost surely.
The convergence of eNn(T ) is ensured under the following assumptions:
(H1) fi is a bounded function for all i = 1, . . . , I.
(H2) lim n→∞Ne
n(0) = y
0 in L1 and almost surely.
(H3) For each compactK of RI, there is a constant CK such that
kF (x) − F (y)k1 ≤ CKkx − yk1
Remark 2 The assumption (H3) can be substituted by the following
as-sumption
(H′3) For each compactK of RI, there is a constant C
′
K such that
kU(x) − U(y)k1 ≤ CKkx − yk1
because of the relation
kF (x) − F (y)k1≤ kU(y)k1kx − yk1+kU(x) − U(y)k1kxk1
Then, the following theorem is a limit theorem on the sequence ( eNn t)
Theorem 3.1 Let assumptions (H1), (H2) and (H3) hold. Then:
lim
n→∞supt≤T k eN n(t)
− ytk1 = 0 in L1
where yt is the unique solution of the equation (9).
3.3 The Central Limit Theorem
Let Xn(t) = √n( eNn(t)− y
t). From the equations (8) and (9), we deduce
that: Xn(T ) = Xn(0) +√n TX−1 t=0 h F ( eNn(t))− F (yt) i +√nMn(T )
We want to show that Xn(T ) converges to the random vector X
T defined by the equation: XT = X0+ TX−1 t=0 dytF.Xt+ MT (10)
where MT is a centered random vector.
The convergence of Xn(T ) is ensured under assumption (H
1) and the following assumptions: (H′′ 2) lim n→+∞X n(0) = X 0 (H′′ 3) F is in class C1
Theorem 3.2 Let assumptions (H1), (H′′2) and (H
′′ 3) hold. Then: lim n→∞supt≤TkX n(t) − Xtk1 = 0 in L1
where Xt verifies the equation (10).
4
Statistical applications
4.1 The likelihood function
Let T > 0 and n∈ N⋆. We suppose that the transitions parameters p
i, qi and
fi for i = 1, . . . , I are entirely determined by a parameter θ taking values in
a set Θ⊂ Rd, with d≥ 1. The likelihood is:
L(n)T (θ) = TY−1 t=0 I Y i=1 Π(n)θ,i (Nn(t), Nn(t + 1))
where, for all m = (m1, . . . , mI) and l = (l1, . . . , lI) Π(n)θ,i(m, li) = min(mXi,li) k=li−min(mi−1,li)
Bθ,ik1 (n)(m, l)Bθ,ik2 (n)(m, l), if li ≤ mi+ mi−1
0 else
and, for i = 1, the convolution of a binomial and a Poisson distributions:
Π(n)θ,1(m, l1) = min(m1,l1) X k=0 Bθ,ik1 (n)(m, l)D(n)θ,k(m, l) with: Bθ,ik1 (n)(m, l) = Ck mip n
θ,i(m)k(1− pnθ,i(m))mi−k for i = 1, . . . , I
Bθ,ik2 (n)(m, l) = Cli−k
mi−1q
n
θ,i−1(m)li−k(1− qθ,i−1n (m))mi−1−li+k for i = 2, . . . , I
Dθ,k(n)(m, l) = PI j=1fθ,jn (m)mj l1−k (l1− k)! e−PIj=1fθ,jn (m)mj Let lTn(θ) = log[L(n)T (θ)].
4.2 The MLE consistency
Consider the ML estimator for T > 0 and n∈ N∗:
ˆ θn= arg max θ∈Θl (n) T (θ). Model identifiability
Let assume that the map θ ∈ Θ 7→ vθ is injective for any vital rate v.
Then, we want to show that for any (θ, θ∗)∈ Θ2:
1 n[l (n) T (θ)− l (n) T (θ ∗)] Pθ∗ −−−→n→∞ K(θ, θ∗) where K(θ, θ∗) = 0 iff θ = θ∗. Proof: For n enough large, Bθ,ik1 (n)(Nn(t), Nn(t + 1)) ∼ Ck nyt,ipθ,i(yt) k(1− p θ,i(yt))nyt,i−k ∼ (nyt,i) k k! pθ,i(yt) 1− pθ,i(yt) k (1− pθ,i(yt))nyt,i
with K1 θ,i(t, k) = 1 k! pθ,i(yt) 1− pθ,i(yt) k
. In the same way: B2 (n)θ,ik (Nn(t), Nn(t + 1)) ∼ K2 θ,i(t, k) ˜B 2 (n) θ,ik (t) with ˜ Bθ,ik2 (n)(t) = 1 (nyt+1,i− k)! qθ,i−1(yt) 1− qθ,i−1(yt) nyt+1,i
(nyt,i−1)nyt+1,i−k(1− qθ,i−1(yt))n(yt,i−1−yt+1,i)
Kθ,i2 (t, k) = 1− qθ,i−1(yt) qθ,i−1(yt) k (1− qθ,i−1(yt))k
4.2.1 MLE asymptotic normality Fisher Information
For all θ, the fisher information matrix is given by: I(θ) = E[∇θlT(θ)]2
where∇θF is the gradient vector of F in θ.
We have: ∇θBθ,ik1 = ∇θpθ,i k− mipθ,i pθ,i(1− pθ,i) Bθ,ik1 for i = 1, . . . , I ∇θBθ,ik2 = ∇θqθ,i
(li− k) − mi−1qθ,i−1
qθ,i(1− qθ,i) Bθ,ik2 for i = 2, . . . , I ∇θDθ,k = I X j=1 ∇θfθ,jmj n1− k PI j=1fθ,j(m)mj − 1 ! Dθ,k Then, for i = 2, . . . , I: ∇θΠθ,t i= min(mXi,li) k=li−min(mi−1,li) B1θ,ikBθ,ik2 ∇θpθ,i k− mipθ,i pθ,i(1− pθ,i) +∇θqθ,i
(li− k) − mi−1qθ,i−1
qθ,i(1− qθ,i) and for i = 1: ∇θΠθ,1 = min(m1,l1) X k=0 B1θ,1kDθ,k ∇θpθ,1 k− m1pθ,1 pθ,1(1− pθ,1) + I X j=1 ∇θfθ,jmj n1− k PI j=1fθ,j(m)mj − 1 !
Appendices
A
Proof of proposition 1
The markov chain (N (t))t∈N is not an irreducible chain, the stage of
hypothesis of "immigration", we can establish a convergence of the process to the stage{0I} with an exponential rate.
Let V the application of NI in [1; +∞[, defined for all N = (N
i)i=1,...,I
by: V (N ) =PIi=1Ni. The drift function at time t related to the function
V , ∆Vt, is defined by the relation:
∆Vt = E(V (N (t + 1))|N(t)) − V (N(t)) = I X i=1 [E(Ni(t + 1)|Nt)− Ni(t)]
From equation (3) given the evolution of the effective vector in mean, we deduce the expression of the drift function:
∆Vt = I X i=1 fi(N (t))Ni(t)− "I−1 X i=1 [1− pi(N (t))− qi(N (t))] Ni(t) + [1− pI(N (t))] NI(t) # = I X i=1 [fi(N (t))− mi(N (t))] Ni(t)
Denote strictly positive real β defined by β = infi=1,...,Imi(N ) and suppose
that it exists constant c > 0 such sup
i=1,...,I
fi(N )kNk1 ≤ c
Then
∆V (N )≤ −βV (N) + c Let now define the markov chain N by:
N = N + L
where L takes values in RI, and is equal to L = (l, 0, . . . , 0) with: l∼ P(λ), λ > 0.
The markov chain N is irreducible, recurrent. Its drift function expresses itself as: ∆V (N ) = ∆V (N ) + ∆V (L) which verifies: ∆V (N ) ≤ −βV (N) + c + λ − V (L) ≤ −β′V (N ) + c′ with β′ and c′ > 0.
B
Proof of proposition 3.1
We first show the following proposition:
Proposition B.1 Let assumptions (H1) and (H2) hold. Then, for all t:
1. Ne n(t) n n→∞ −−−→ 0 in L1. 2. eNn(t) is square integrable. 3. V( eNn(t + 1)|Fn t) n→∞ −−−→ 0 in L1. Proof: 1. As eNn(0) tends to y
0 in L1, eNn(0) verifies the property. Now, the
equation (1) of the model can be also written like this: Nin(t + 1) = Nin(t) + Fi−1 in (t)− (Fn
i i+1(t) + Fin†(t))
Therefore by summing we obtain:
kNn(t + 1)k1 =kNn(t)k1+ Rn(t)− Mn(t)
where Mn(t) =PI
i=1Fin†(t)) is the total number of dead trees between
t and t + 1. We deduce the inequality
kNn(t + 1)k1 ≤ Rn(t) +kNn(t)k1 (11) implying E k eN n(t + 1)k 1 n ! ≤ E Rn(t) n2 + E k eN n(t)k 1 n ! ≤ 1 n I X i=1 E(fθ,i( eNn(t)) eNin(t)) + E k eN n(t)k 1 n !
Now, by the assumption (H1), it exists a real A > 0 such as:
E k eN n(t + 1)k 1 n ! ≤ (1 + A) E k eN n(t)k 1 n !
as if to this relation for t = 0, . . . , T : E k eN n(t)k 1 n ! ≤ (2 + A)tE k eNn(0)k1 n ! Then, as k eN n(0)k 1 n tends to zero in L 1, we deduce that k eNn(t)k1 n tends also to zero in L1
2. From the equation (11), we deduce that: E(k eNn(t + 1)k21) ≤ E Rn(t)2 n2 + 4E Rn(t) n k eN n(t) k1 + 4 Ek eNn(t)k21 ≤ I X i=1 E(fθ,i( eNn(t)) eNin(t)) . . . ≤ 2 +A n 2 E(k eNn(t)k21) +A nE(k eN n(t)k 1)
As eNn(0) is square integrable, we prove by recurrence that eNn(t) is
also like this for all t. 3. Now, V( eNn(t)|Fn
t) =
1 n2V
n(t), and the expression of Vn(t) is given
by the equation (??). As pn
θ,i(Nn(t)) + qnθ,i(Nn(t)) ≤ 1 for all n and
t, and as Ne n(t) n tends to zero in L 1, therefore V( eNn(t)|Fn t ) tends to zero in L1.
Lemma 1 Let assumptions (H1) and (H2) hold. Then:
lim
n→∞supt≤T kM n(t)k
1= 0 in L2 and almost surely
Proof
Step 1. We prove the convergence of supt≤T kMn(t)k
1 in L2.
• First, we verify that Mn(t) is square integrable. Indeed :
E kMn(t)k 12 ≤ t−1 X s=0 E k eNn(s + 1)− ENen(s + 1)|Fn s k1 2 ≤ 2 t−1 X s=0 E h k eNn(s + 1)k1 2i
We conclude by the proposition B.1. • Let Mn
i (t) the ith composant of Mn(t), for i = 1, . . . , I. The
series Mn i (t) is in L2 and: E[Min(t)2] = E[(Min(t)− Min(0))2] = E t−1 X s=0 (Min(s + 1)− Mn i (s)) !2
As Mn
i (t) is a martingale, we deduce that:
E[Min(t)2] = E "t−1 X s=0 (Min(s + 1)− Min(s))2 # = t−1 X s=0 E[V( eNin(s + 1)|Fsn)]
By the proposition B.1 we deduce that Mn
i (t) converges to zero
in L2.
• Now, from the Doob’s inequality: E[(sup t≤T kM n(t) k1)2] ≤ 4E[kMn(T )k21] ≤ 4E I X i=1 |Mn i (T )| !2 ≤ 4 I X i=1 E[Min(T )2]
Then, we deduce that supt≤T kMn(t)k
1 tends to zero in L2.
Step 2. [as convergence] Let Mn
∗ = supt≤TkMn(t)k1. M∗nconverges to zero in
L2 so in probability. Then, by the Borel-Cantelli’s lemma, it exists an
under-series of (Mn
∗)n which converges to zero almost surely. We have
to show that (Mn
∗)n is a Cauchy series almost surely.(. . .)
Proof of theorem 3.1: First, as the sequence ( eNn(t)), for t = 0, . . . , T ,
are bounded in L1, it exists a compact K in RI,k.k L1((µ
0,Π),k.k1)
which contains {y0, . . . , yT} and { eNn(0), . . . , eNn(T )} for all n, where:
kfkL1(µ,k.k 1)=
Z
kfk1dµ
for all measurable function f with values in RI.
On the other hand, for all t≤ T and in L1:
k eNn(t)− ytk1 = k eNn(0)− y0+ Mn(t) + t−1 X s=0 F ( eNn(s)− F (ys) k1 ≤ k eNn(0)− y0k1+kMn(t)k1+ t−1 X s=0 kF ( eNn(s))− F (ys)k1 ≤ k eNn(0)− y 0k1+kMn(t)k1+ t−1 X s=0 CKk eNn(s)− ysk1
By Gronwall’s inequality this implies that: k eNn(t)− ytk1 ≤ k eNn(0)− y0k1+kMn(t)k1 eCKt
Then, taking the supremum on t, we obtain: sup t≤T k e Nn(t)− ytk1 ≤ k eNn(0)− y0k1+ sup t≤T kM n(t)k 1 ! eCKT
The first term in the brackets converges to 0 in L1 by assumption, and the second term by the lemma 1. The exponential function is moreover independent of n. This completes the proof of the theorem.
C
Proof of theorem 3.2
Let Γ(t) defined , for i = 2, . . . , I, by:
Γii(t) = yt,ipi(yt)[1− pi(yt)] + yt,i−1qi−1(yt)[1− qi−1(yt)]
Γi−1i(t) = Γtii−1 =−yt,i−1pi−1(yt)qi−1(yt) (12)
Γ11(t) = yt,1p1(yt)[1− p1(yt)] + I
X
i=1
fi(yt)yt,i
Proposition C.1 Let assumptions (H1), (H2) and (H
′
3) hold. Then, for all
t
nV( eNn(t + 1)|Ftn)−−−→ Γ(t)n→∞ in probability
Proof The proposition is a straight result from the theorem 3.1 and from the continuity of U .
Lemma 2 It exists a real c such as for all t = 1, . . . , T E(kXn(t)k1) < c
Proof Let t = 1, . . . , T . The random vector Xn(t) verifies
Xn(t) = Xn(0) +√n t−1 X s=0 h F ( eNn(s))− F (ys) i +√nMn(t)
Step 1. From the definition of the function F F ( eNn(s))− F (y s) = h U ( eNn(s)) eNn(s)− U(y s)ys i + [ys− eNn(s)] = hU ( eNn(s))− U(ys) i e Nn(s) + (1− U(ys))[ys− eNn(s)]
Then, from the assumption (H3′) √ nkF ( eNn(s))− F (y s)k ≤ h CK′ Nen(s) + (1− U(y s)) i kXn(s)k
Step 2. As the same as the step 1 of the proof of the lemma 1, E[kMn(t)k21]≤ I X i=1 t−1 X s=0 E[V( eNin(s + 1)|Fsn)] Then (√nE[kMn(t)k1])2 ≤ I X i=1 t−1 X s=0 nE[V( eNin(s + 1)|Fn s)]
By the proposition C.1,√nE[kMn(t)k1] is bounded.
Step 3. Hypothesis (H′′
2) unsures that it exists a real c0such that E(kXn(0)k) < c0.
By a recurrence we deduce the lemma.
Lemma 3 Let assumptions (H1), (H2) and (H
′
3) hold. Then:
√
nMn(T )−−−→ N (0, Γ)n→∞ in law
where Γ =PTt=0−1Γ(t).
Proof Let Φn the characteristic function of √nMn(T ) and N n (t) = √ nhNen(t)− E( eNn(t)|Fn t−1) i
for t = 1, . . . , T . For all λ∈ RI
Φn(λ) = E exp ihλ,√nMn(T )i √ n = TY−1 t=0 Eexp ihλ, Nn(t)i√n = TY−1 t=0 1−1 2 tλV(Nn(t))λ +◦ V(Nn(t)) = exp[−1 2 TX−1 t=0 tλV(Nn(t))λ]
From the proposition C.1 we deduce that Φn(λ)−−−→ exp[−n→∞
1 2
tλΓλ]
Proof of theorem 3.2: The Taylor’s development of F gives: √ nF ( eNn(t))− F (yt) = √ndytF.( eN n(t)− y t) + O(√nk eNn(t)− ytk21) = dytF.X n(t) + O( k eNn(t)− ytk1)Xn(t)
Then for all t≥ 1 Xn(t) = X0n+ t−1 X s=0 dysF.X n(s) + O( k eNn(s)− ytk1)Xn(s) +√nMn(t) and Xn(t)− Xt= (X0n− X0) + t−1 X s=0 dysF.(X n(s) − Xs) + ( √ nMn(t)− Mt) + ǫn(t)
where Xtverifies the equation (10) and ǫn(t) =Pt−1s=0O(k eNn(s)−ytk1)Xn(s).
As the same as the proof of the theorem 3.1, we deduce by Gronwall inequality that sup t≤TkX n(t)−X tk1 ≤ (kX0n−X0k1+sup t≤T k √ nMn(t)−Mtk1)eCFT+sup t≤T kǫn (t)k1 where CF =kdF k∞.
The first term in the brackets converges to 0 in L1 by assumption and the second term by the lemma 3. By theorem 3.1 and lemma 2 we deduce the convergence of supt≤T kǫn(t)k1toward to zero. The exponential function
being independent of n this completes the proof of the theorem.
References
[1] E.J. Allen, L.J.S Allen, and H. Schurz. A comparison of persistence-time estimation for discrete and continuous stochastic population mod-els that include demographic and environmental variability. Mathemat-ical Biosciences, 196:14–38, 2005.
[2] L.J.S Allen. An Introduction to Stochastic Processes with Applications to Biology. Prentice Hall, Upper Saddle River, NJ, 2003.
[3] E. R. Alvarez-Buylla, R. García-Barrios, C. Lara-Moreno, and M. Martínez-Ramos. Demographic and genetic models in conservation biology: applications and perpectives for tropical rain forest tree species. Annual Review of Ecology and Systematics, 27:387–421, 1996.
[4] M. Boscolo and J. R. Vincent. Promoting better logging practices in tropical forests: A simulation analysis of alternative regulations. De-velopment Discussion Paper 652, The Harvard Institute for Interna-tional Development, Harvard University, Cambridge, Massachusetts, USA, 1998.
[5] M. S. Boyce. Population viability analysis. Annual Review of Ecology and Systematics, 23:481–506, 1992.
[6] J. Buongiorno, S. Dahir, H. C. Lu, and C. R. Lin. Tree size diversity and economic returns in uneven-aged forest stands. Forest Science, 40(1):83–103, 1994.
[7] J. Buongiorno and J. K. Gilless. Decision Methods for Forest Resource Management. Academic Press, 2003.
[8] J. Buongiorno, J. L. Peyron, F. Houllier, and M. Bruciamacchie. Growth and management of mixed-species, uneven-aged forests in the French Jura: implications for economic returns and tree diversity. Forest Sci-ence, 41(3):397–429, 1995.
[9] H. Caswell. Matrix Population Models: Construction, Analysis and Interpretation. Sinauer Associates, Inc. Publishers, Sunderland, Mas-sachusetts, 2 edition, 2001.
[10] J. M. Cushing. Matrix Models and Population Dynamics. Mathematics Series. American Mathematical Society, IAS/Park City, james keener and mark lewis edition, 2008.
[11] J. M. Cushing, R.F. Constantio, B. Dennis, R.A. Desharnais, and S. M. Henson. Chaos in Ecology. Academic Press, Amsterdam, Boston, Lon-don, New-York, 2003.
[12] B. Dennis and M.L. Taper. Density dependence in time series observa-tions of natural populaobserva-tions: estimation and testing. Ecological Mono-graph, 64:205–224, 1994.
[13] W. G. Doubleday. Harvesting in matrix population models. Biometrics, 31(1):189–200, 1975.
[14] V. Favrichon. Modeling the dynamics and species composition of tropi-cal mixed-species uneven-aged natural forest: Effects of alternative cut-ting regimes. Forest Science, 44(1):113–124, 1998.
[15] Suresh Kumar Ghimire, Olivier Gimenez, Roger Pradel, Doyle McKey, and Yildiz Aumeeruddy-Thomas. Blackwell publishing ltd demographic variation and population viability in a threatened himalayan medicinal and aromatic herb nardostachys grandiflora : matrix modelling of har-vesting effects in two contrasting habitats. Journal of Applied Ecology, 45:41–51, 2008.
[16] C. E. Hauser, E. G. Cooch, and J. D. Lebreton. Control of structured populations by harvest. Ecological Modelling, 196(3-4):462–470, 2006. [17] D. Ingram and J. Buongiorno. Income and diversity tradeoffs from
management of mixed lowland dipterocarps in Malaysia. Journal of Tropical Forest Science, 9(2):242–270, 1996.
[18] M. Jarry, P. Davaine, and E. Beall. A matrix model to study the colo-nization by brown trout of a virgin ecosystem in the Kerguelen islands. Acta Biotheoretica, 46(3):253–272, 1998.
[19] W. R. Lamar and J. B. McGraw. Evaluating the use of remotely sensed data in matrix population modeling for eastern hemlock (Tsuga canadensis L.). Forest Ecology and Management, 212(1-3):50–64, 2005. [20] R. Lande. Risks of population extinction from demographic and envi-ronmental stochasticity and random catastrophes. The American Nat-uralist, 142:911–927, 1993.
[21] L. P. Lefkovitch. The study of population growth in organisms grouped by stages. Biometrics, 21:1–18, 1965.
[22] P. H. Leslie. In the use of matrices in certain population mathematics. Biometrika, 33:183–212, 1945.
[23] J. Liang, J. Buongiorno, and R. A. Monserud. Growth and yield of all-aged douglas-fir – western hemlock forest stands: a matrix model with stand diversity effects. Canadian Journal of Forest Research, 35(10):2368–2381, 2005.
[24] C. R. Lin, J. Buongiorno, and M. Vasievich. A multi-species, density-dependent matrix growth model to predict tree diversity and income in northern hardwood stands. Ecological Modelling, 91(1-3):193–211, 1996. [25] H. C. Lu and J. Buongiorno. Long- and short-term effects of alternative cutting regimes on economic returns and ecological diversity in mixed-species forests. Forest Ecology and Management, 58(3-4):173–192, 1993. [26] E. S. Menges. Population viability analyses in plants: challenges and
opportunities. Trends in Ecology and Evolution, 15(2):51–56, 2000. [27] B. R. Michie and J. Buongiorno. Estimation of a matrix model of
forest growth from re-measured permanent plots. Forest Ecology and Management, 8:127–135, 1984.
[28] W. F. Morris and D. F. Doak. Quantitative Conservation Biology: The-ory and Practice of Population Viability Analysis. Sinauer Associates, Inc., Sunderland, MA, 2002.
[29] M. G. Neubert and H. Caswell. Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations. Ecology, 81(6):1613–1628, 2000.
[30] Y. Takeuchi, Y. Iwasa, and K. Sato. Mathematics for Ecology and Environmental Sciences, volume 1 of Biological and Medical Physics, Biomedical Engineering. Srpinger, Berlin, 2007.
[31] M. B. Usher. A matrix approach to the management of renewable re-sources, with special reference to the selection forests. Journal of Applied Ecology, 3:355–367, 1966.
[32] M. B. Usher. The relation between mean square and block size in the analysis of similar patterns. Journal of Ecology, 57:505–514, 1969. [33] J. K. Vanclay. Growth models for tropical forests: A synthesis of models
